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Ever-growing edge applications often require short processing latency and high energy efficiency to meet strict timing and power budget. In this work, we propose that the compact long short-term memory (LSTM) model can approximate conventional acausal algorithms with reduced latency and improved efficiency for real-time causal prediction, especially for the neural signal processing in closed-loop feedback applications. We design an LSTM inference accelerator by taking advantage of the fine-grained parallelism and pipelined feedforward and recurrent updates. We also propose a bit-sparse quantization method that can reduce the circuit area and power consumption by replacing the multipliers with the bit-shift operators. We explore different combinations of pruning and quantization methods for energy-efficient LSTM inference on datasets collected from the electroencephalogram (EEG) and calcium image processing applications. Evaluation results show that our proposed LSTM inference accelerator can achieve 1.19 GOPS/mW energy efficiency. The LSTM accelerator with 2-sbit/16-bit sparse quantization and 60% sparsity can reduce the circuit area and power consumption by 54.1% and 56.3%, respectively, compared with a 16-bit baseline implementation.